Convergence Theorems and Convergence Rates for the General Inertial Krasnosel’skiǐ–Mann Algorithm

The authors [13] introduced a general inertial Krasnosel’skiǐ–Mann algorithm: $$ \left\{ \begin{aligned}&y_n=x_n+\alpha _n(x_n-x_{n-1}),\\&z_n=x_n+\beta _n(x_n-x_{n-1}),\\&x_{n+1}=(1-\lambda _n)y_n+\lambda _nT(z_n) \end{aligned} \right. $$ y n = x n + α n ( x n - x n - 1 ) , z n = x n + β n ( x n - x n - 1 ) , x n + 1 = ( 1 - λ n ) y n + λ n T ( z n ) for each $$n\ge 1$$ n ≥ 1 and showed its convergence with the control conditions $$\alpha _n,\beta _n\in [0,1).$$ α n , β n ∈ [ 0 , 1 ) . In this paper, we present the convergence analysis of the general inertial Krasnosel’skiǐ–Mann algorithm with the control conditions $$\alpha _n\in [0,1]$$ α n ∈ [ 0 , 1 ] , $$\beta _n\in (-\infty ,0]$$ β n ∈ ( - ∞ , 0 ] and $$\alpha _n\in [-1,0]$$ α n ∈ [ - 1 , 0 ] , $$\beta _n\in [0,+\infty )$$ β n ∈ [ 0 , + ∞ ) , respectively. Also, we provide the convergence rate for the general inertial Krasnosel’skiǐ–Mann algorithm under mild conditions on the inertial parameters and some conditions on the relaxation parameters, respectively. Finally, we show that a numerical experiment provided compares the choice of inertial parameters.